The Greeks and hedging

The option P&L equation

Attributes a day’s profit to direction, convexity, decay and repricing — and answers most Greek questions if you read it carefully.

dΠ≈Δ dS+12Γ (dS)2+Θ dt+ν dσ\mathrm{d}\Pi \approx \Delta\,\mathrm{d}S + \tfrac{1}{2}\Gamma\,(\mathrm{d}S)^2 + \Theta\,\mathrm{d}t + \nu\,\mathrm{d}\sigma

Where

Δ dS\Delta\,\mathrm{d}S
Directional, and removed by delta hedging.
12Γ(dS)2\tfrac{1}{2}\Gamma(\mathrm{d}S)^2
Positive when long gamma, whichever way the move goes.
Θ dt\Theta\,\mathrm{d}t
The rent paid for that convexity.
ν dσ\nu\,\mathrm{d}\sigma
Profit from a repricing of implied volatility.

Assumptions

  • A second-order expansion, so it degrades on very large moves.

Sanity check. Delta-hedge and the first term vanishes, leaving a bet that realised volatility beats implied.

Where this is taught